In Ref. [1]}, a QME called the ULE has been derived,
which preserves the positivity of the reduced dynamics.
The derivation does not rely on the secular approximation and has the same level of accuracy as the Redfield equation.
In this section, we calculate the perturbative steady state of the ULE.
We focus on the case of the time-independent system Hamiltonian, in which case the ULE takes the form in the Schrödinger picture
\(\partial _t \rho (t)=\mathcal {L}^{(0)}[\rho (t)]+\epsilon ^2 \mathcal {L}^{(2)}_{\rm ULE}[\rho (t)]\)  [1]}, where
the \(O(\epsilon ^2)\)  superoperator has two components: \(\mathcal {L}^{(2)}_{\rm ULE}=\mathcal {L}^{(a)}_{\rm ULE}+\mathcal {L}^{(b)}_{\rm ULE}\) . The first one takes the form similar to the Lamb-shift term as
\(\mathcal {L}^{(a)}_{\rm ULE}[\rho (t)]=-i[\Lambda ,\rho (t)]\)  with
\(\Lambda &= \sum _{\alpha ,\beta } \sum _{\omega ,\omega ^\prime } f_{\alpha \beta } (\omega ,\omega ^\prime ) A_\alpha (\omega ) A_\beta (\omega ^\prime ),\) 
